Utilix
Math & numbers·Geometry

Pythagorean Theorem Solver

Solve any right triangle with the Pythagorean theorem. Enter two sides and get the third side instantly, with full step-by-step working and triple detection.

Input

Result

Enter a value for solve for to see your result.

How it works

Solves a right triangle with the Pythagorean theorem (a² + b² = c²). Enter two known sides and the tool computes the third, with the full step-by-step working shown.

Formula

a² + b² = c²

a
Length of one leg of the right triangle
b
Length of the other leg
c
Length of the hypotenuse (opposite the right angle)

Step by step

  1. 01Decide which side you need to solve for: the hypotenuse c, or one of the legs.
  2. 02If you are solving for the hypotenuse: c = √(a² + b²).
  3. 03If you are solving for a leg: rearrange to leg = √(c² − a²); the hypotenuse must be longer than the known leg.
  4. 04Square the known sides, add or subtract per the rearranged formula, then take the square root.
  5. 05If the inputs form a known integer triple (3-4-5, 5-12-13, 8-15-17, …) the tool flags it as a Pythagorean triple.

Examples

3-4-5 triangle

c = √(3² + 4²) = √(9 + 16) = √25 = 5. The classic 3-4-5 right triangle.

Inputs

Solve for:
hypotenuse
Leg a:
3
Leg b:
4

Result

Missing side:
5

Solve for a leg (13² − 5² → 12)

leg = √(13² − 5²) = √(169 − 25) = √144 = 12. The 5-12-13 Pythagorean triple.

Inputs

Solve for:
leg
Leg a:
5
Hypotenuse c:
13

Result

Missing side:
12
Note: Hypotenuse c must be strictly greater than each leg. If c ≤ a in 'solve for leg' mode, the tool returns a friendly message instead of NaN. Negative side lengths are not allowed; clamp to 0 if you accidentally type a minus sign.

Frequently asked questions

What is the Pythagorean theorem?

In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It only applies when one angle is exactly 90°.

How do I solve for a leg instead of the hypotenuse?

Rearrange the formula: a = √(c² − b²). The hypotenuse must be longer than the leg you already know, otherwise the square root is of a negative number and there is no real solution.

What is a Pythagorean triple?

A set of three positive integers (a, b, c) that satisfy a² + b² = c². Common triples are 3-4-5, 5-12-13, 8-15-17, and 7-24-25. The tool flags inputs that form one.